Free Zinbiel algebras were shown to
be precisely the shuffle product algebra [1]}.
Naurazbekova proved that, over a field of characteristic zero, free Zinbiel algebras are the free associative-commutative algebras (without unity) with respect to the symmetrization multiplication and their free generators are found; also she constructed examples of subalgebras of the two-generated free Zinbiel algebra that are free Zinbiel algebras of countable rank [2]}.
Nilpotent algebras play an important role in the class of Zinbiel algebras.
So,
Dzhumadildaev and Tulenbaev proved that each complex finite dimensional Zinbiel algebra is nilpotent [3]};
Naurazbekova
and Umirbaev proved that in characteristic zero any proper subvariety of the variety of Zinbiel algebras is nilpotent [4]}.
Finite-dimensional Zinbiel algebras with a “big” nilpotency index are classified in [5]}, [6]}.
Central extensions of three dimensional Zinbiel algebras were calculated in [7]} and of filiform Zinbiel algebras in [8]}.
The full system of degenerations of complex four dimensional Zinbiel algebras is given in [9]}.
